Solve x=x^2-30x+125

$$x = { x }^{ 2 } -30x+125$$

Solve for x

$$x = \frac{\sqrt{461} + 31}{2} \approx 26.235455277$$

$$x = \frac{31 - \sqrt{461}}{2} \approx 4.764544723$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x-x^{2}=-30x+125$$

Add 30x to both sides.

$$x-x^{2}+30x=125$$

Combine x and 30x to get 31x.

$$31x-x^{2}=125$$

Subtract 125 from both sides.

$$31x-x^{2}-125=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+31x-125=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 31 for b, and -125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-1\right)\left(-125\right)}}{2\left(-1\right)}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\left(-1\right)\left(-125\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-31±\sqrt{961+4\left(-125\right)}}{2\left(-1\right)}$$

Multiply 4 times -125.

$$x=\frac{-31±\sqrt{961-500}}{2\left(-1\right)}$$

Add 961 to -500.

$$x=\frac{-31±\sqrt{461}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-31±\sqrt{461}}{-2}$$

Now solve the equation x=\frac{-31±\sqrt{461}}{-2} when ± is plus. Add -31 to \sqrt{461}.

$$x=\frac{\sqrt{461}-31}{-2}$$

Divide -31+\sqrt{461} by -2.

$$x=\frac{31-\sqrt{461}}{2}$$

Now solve the equation x=\frac{-31±\sqrt{461}}{-2} when ± is minus. Subtract \sqrt{461} from -31.

$$x=\frac{-\sqrt{461}-31}{-2}$$

Divide -31-\sqrt{461} by -2.

$$x=\frac{\sqrt{461}+31}{2}$$

The equation is now solved.

$$x=\frac{31-\sqrt{461}}{2} x=\frac{\sqrt{461}+31}{2}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$x-x^{2}=-30x+125$$

Add 30x to both sides.

$$x-x^{2}+30x=125$$

Combine x and 30x to get 31x.

$$31x-x^{2}=125$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+31x=125$$

Divide both sides by -1.

$$\frac{-x^{2}+31x}{-1}=\frac{125}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{31}{-1}x=\frac{125}{-1}$$

Divide 31 by -1.

$$x^{2}-31x=\frac{125}{-1}$$

Divide 125 by -1.

$$x^{2}-31x=-125$$

Divide -31, the coefficient of the x term, by 2 to get -\frac{31}{2}. Then add the square of -\frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-31x+\left(-\frac{31}{2}\right)^{2}=-125+\left(-\frac{31}{2}\right)^{2}$$

Square -\frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-31x+\frac{961}{4}=-125+\frac{961}{4}$$

Add -125 to \frac{961}{4}.

$$x^{2}-31x+\frac{961}{4}=\frac{461}{4}$$

Factor x^{2}-31x+\frac{961}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{2}\right)^{2}=\frac{461}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{2}\right)^{2}}=\sqrt{\frac{461}{4}}$$

Simplify.

$$x-\frac{31}{2}=\frac{\sqrt{461}}{2} x-\frac{31}{2}=-\frac{\sqrt{461}}{2}$$

Add \frac{31}{2} to both sides of the equation.

$$x=\frac{\sqrt{461}+31}{2} x=\frac{31-\sqrt{461}}{2}$$